Advances in Operator Theory
- Adv. Oper. Theory
- Volume 4, Number 1 (2019), 251-264.
Dominated orthogonally additive operators in lattice-normed spaces
In this paper, we introduce a new class of operators in lattice-normed spaces. We say that an orthogonally additive operator $T$ from a lattice-normed space $(V,E)$ to a lattice-normed space $(W,F)$ is dominated, if there exists a positive orthogonally additive operator $S$ from $E$ to $F$ such that $\vert Tx \vert \leq S \vert x \vert$ for any element $x$ of $(V,E)$. We show that under some mild conditions, a dominated orthogonally additive operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated orthogonally additive operator. In the last part of the paper we consider laterally-to-order continuous operators. We prove that a dominated orthogonally additive operator is laterally-to-order continuous if and only if the same is its exact dominant.
Adv. Oper. Theory, Volume 4, Number 1 (2019), 251-264.
Received: 27 April 2018
Accepted: 2 August 2018
First available in Project Euclid: 20 September 2018
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Abasov, Nariman; Pliev, Marat. Dominated orthogonally additive operators in lattice-normed spaces. Adv. Oper. Theory 4 (2019), no. 1, 251--264. doi:10.15352/aot.1804-1354. https://projecteuclid.org/euclid.aot/1537408975